Addition and subtraction are the inverse operations of each other. But this means that they are the opposite. You can undo an addition through subtraction, and you can undo a subtraction through addition.
It is the basis of their relationship, but there’s more to it than that. Let’s start with a refresher on what they are, then look at how addition and subtraction are related and some teaching resources for this topic.
What is addition?
The addition is one of the four Maths operations. It’s where you put numbers or amounts together to find the total they make. For example, an addition sum might be written out like 2 + 5 = 7, or with words, such as two plus five equals seven.
Sometimes, addition forms part of other calculations. For example, if we’re dealing with big numbers, we might need to split the problem and add each part together to get the final answer.
Counting up is the earliest form of addition; if we count forwards from zero to 20, we will keep adding one each time. We add these numbers at every step when we count up in twos, fives, and tens. It may feel more like following a pattern than doing mini-addition sums, but it’s both!
What is subtraction?
Subtraction is also one of the four operations in Maths. This time, instead of bringing things together, we remove some. How many would be left if you had five beads and someone took two of them away? Three, and that’s a subtraction. It would be written mathematically like this, 5 – 2 = 3.
The order in which the numbers appear is essential for subtraction. The number before the minus sign is the amount you start with; in this case, it would be five. The number after the minus sign is the amount you’re taking away, which is two in this example. If we swap them around, we’d have to take five away from two: 2 – 5 = -3. This answer is now different, and we’ve ventured into negative numbers. Always pay attention to the order in subtraction to ensure you’re doing the right calculation and looking for the correct answer.
Counting down is the earliest form of subtraction. When counting down in ones, we take one away every time.
How are addition and subtraction related?
You might feel how addition and subtraction are related just from their definitions. For example, the addition symbol tells us to bring more, whereas the subtraction symbol tells us to do the opposite and take some away.
It’s best to relate this to real objects instead of abstract numbers. Find seven of something, like beads, sweets, socks, or apples. Put five on the table, then add two. There will be seven. You must subtract two to get back down to five apples. You’ll have three on the table if you take away a further two. To get back up to five, add two. We can do the exact opposite to undo the action every time we add or subtract. This visual and tactile activity is a great way for children to explore how addition and subtraction are related, and it’s why many choose to use an abacus for addition and subtraction.
Through this experimentation, we’ve found that there’s not only a strong relationship between addition and subtraction themselves but also between specific problems. So, for example, we know that 5 + 2 = 7 and that 7 – 2 = 5. These are facts, and we can group them into fact families.
What are fact families?
They are groups of short equations that use the same three numbers. So they’ll always be true, which is why they’re facts we can rely on for future calculations.
So what are the fact families we’ve looked at so far? We’ve found a fact family for five, two, and seven. Here are the four facts we can make with them:
- 5 + 2 = 7
- 2 + 5 = 7 (order isn’t a problem for addition, so we can swap the first two numbers without affecting the answer)
- 7 – 2 = 5
- 7 – 5 = 2
Can you notice how there are pairs of opposites? Swapping the first number and the answer in each addition makes a correct subtraction and vice versa. You can see this even more clearer in a table. So, let’s look at the fact family for the numbers two, three, and five, which we also experimented with.
3 + 2 = 5
5 – 2 = 3
2 + 3 = 5
5 – 3 = 2
They are mirror images of each other with the symbols changed; this is an example of the beautiful patterns we find occurring naturally in Maths.
Knowing how this works, can you write the fact family for four, six, and ten?
Here it is for you to check your answers:
4 + 6 = 10
10 – 6 = 4
6 + 4 = 10
10 – 4 = 6
How do we use fact families?
As well as showing pretty patterns, fact families are useful for math calculations. For example, suppose children are comfortable with how addition and subtraction are related and how sets of three numbers are related by addition and subtraction. In that case, they can complete problems much more quickly. They can recognize which numbers go together without counting out the sum, and subtraction feels much less scary when we understand it as the reverse of addition.
Opposites of the same methods
In Maths, we use lots of different strategies for completing operations. But, because of how addition and subtraction are related, we can easily use some of the same methods. But we have to do the opposite.
The jump strategy, also known as counting on, is a great way to learn how to do simple addition and subtraction. On a number line, you start at the first number in your calculation, then jump the right amount of spaces for the second number. The place where you land is the answer.
Let’s say that, once again, we’re looking at 5 + 2. Begin at five, then take two jumps to the right, going up the number line. One jump takes you to six, and two jumps take you to seven.
For subtraction, we do the opposite. So, for 7 – 2, start at seven and take two jumps to the left, going down the number line. One jump takes you to six, and two jumps back to five. With bigger numbers and more confidence, children can also make larger jumps of ten at a time. You can easily find your answer if you know which way to go for the operation you’re doing, and you jump the right amount of space.
However, this method would take far too long when the numbers get much bigger. A more advanced strategy that we use for both addition and subtraction is the column method.
It is where you line up your numbers based on their place value and complete smaller addition or subtraction problems for each column. Because these operations are so similar, this layout works for them both. Remember which calculation you are doing; don’t add some columns and subtract others.
How are addition and subtraction related to multiplication and division?
Multiplication and division are the other pair of opposite Maths operations. Early on, children will look at these about addition and subtraction. They’ll be taught to see multiplication as repeated addition and division as repeated subtraction.
Here are some simple examples. The multiplication 3 × 4 can also be seen as four lots of three being added together from zero:
- 0 + 3 = 3
- 3 + 3 = 6
- 6 + 3 = 9
- 9 + 3 = 12
We can also lay it out as 3 + 3 + 3 + 3 = 12. Let’s take another example, 6 × 5 is just a way of saying we add six together five times: 6 + 6 + 6 + 6 + 6 = 30.
In division, we take away a number; however, often, we need to get down to zero. We don’t know how many it will take, so that’s the answer we’re looking for. If we want to do 12 ÷ 3, we keep taking away three:
- 12 – 3 = 9
- 9 – 3 = 6
- 6 – 3 = 3
- 3 – 3 = 0
We took three away from 12 four times, so we know that 12 ÷ 3 = 4. It is the opposite of what we did above. The inverse relationship between addition and subtraction means we use a different one depending on whether it’s multiplication or division. Still, in both cases, we repeat our calculation, keep track of how many times we do so and use this process to find our answer.